Optimality Regions and Fluctuations for Bernoulli Last Passage Models

TL;DR

This paper investigates the structure of optimality regions in Bernoulli last passage models and identifies conditions under which the last passage time follows a Tracy-Widom distribution, advancing understanding of these stochastic processes.

Contribution

It provides rigorous bounds on the number of optimality regions and characterizes the convergence to Tracy-Widom law in the independent model at zero penalty.

Findings

Upper bounds for optimality regions near the soft edge

Exact solvability of the independent model at zero penalty

Convergence of last passage time to Tracy-Widom law

Abstract

We study the sequence alignment problem and its independent version, the discrete Hammersley process with an exploration penalty. We obtain rigorous upper bounds for the number of optimality regions in both models near the soft edge. At zero penalty the independent model becomes an exactly solvable model and we identify cases for which the law of the last passage time converges to a Tracy-Widom law.